Method and apparatus for examining features on semi-transparent and transparent substrates

ABSTRACT

An apparatus and method for determining a physical parameter of features on a substrate by illuminating the substrate with an incident light covering an incident wavelength range Δλ, e.g., from 190 nm to 1000 nm, where the substrate is at least semi-transparent. A response light received from the substrate and the feature is measured to obtain a response spectrum of the response light. Further, a complex-valued response due to the feature and the substrate is computed and both the response spectrum and the complex-valued response are used in determining the physical parameter. The response light is reflected light, transmitted light or a combination of the two. The complex-valued response typically includes a complex reflectance amplitude, a complex transmittance amplitude or both. The apparatus and method take into account the effects of vertical and lateral coherence length and are well suited for examining adjacent features.

FIELD OF THE INVENTION

[0001] The present invention relates generally to methods and apparatusfor optically examining features on semi-transparent substrates andespecially to examining a number of adjacent features on suchsemi-transparent substrates.

BACKGROUND OF THE INVENTION

[0002] In the production of miniaturized objects such as miniaturedevices including integrated circuits and microelectronics forsemiconductor and display applications, the tools and auxiliarystructures used in their manufacture, as well as the miniature objectsthemselves have to be examined carefully. Optical methods of examiningthese tools and objects are non-destructive and frequently preferredover other approaches. Hence, advances in optical examination ofminiature features including patterns composed of adjacent features areimportant.

[0003] In many cases miniature devices are made by photolithographictechniques. In a typical application of the photolithographic technique,a layer of photoresist is deposited on a substrate or other device layerand then exposed to radiation of appropriate wavelength through apatterning mask. Of course, the masks themselves also need to beappropriately patterned with miniature features to be able to performtheir function and are thus themselves a class of miniature devices thathas to be examined.

[0004] Now in photolithography certain regions of the photoresist layerare exposed and others are not, according to the pattern defined in thepatterning mask. Exposing the photoresist to radiation changes itssolubility. After exposure, solvent is used to remove regions of highersolubility photoresist, leaving regions of “hardened” photoresist atsites on the device layer as dictated by the patterning mask. The“hardened” photoresist remains to protect the underlying material fromremoval during a subsequent etching step or other suitable materialremoval procedure. After etching the photoresist is discarded. In thismanner, a feature is created in the device based on the pattern definedin the mask.

[0005] Clearly, the photoresist layer must be accurately patterned toform features to the exacting specifications for miniature devices. Itis therefore desirable to monitor the photolithographic process atvarious stages and on a periodic basis. For example, it would bedesirable to measure the thickness of the photoresist layer and examinethe pattern to determine feature sizes. The thickness can be measured bysubjecting the photoresist to light with a wavelength in the range of190 to 1000 nm and measuring the reflected light. The reflectedradiation may be correlated to photoresist thickness. The generalprinciple of this measurement technique is that the measured lightreflected from a substrate is modulated by constructive and destructiveoptical interference from an overlaying semitransparent material such asthe photoresist. For more information see Chopra, K. L., Thin FilmPhenomena, p. 99 (McGraw Hill, 1969). The periodicity of the reflectancespectra can also be used to determine optical properties, such as therefractive index n of the substrate.

[0006] Measurement of the pattern or features is a more difficultprocedure. For example, in a typical application, the pattern consistsof a plurality of stripes and spaces, e.g., a line and space pattern.These types of patterns are frequently encountered in formingdiffractive elements such as lenses or gratings in semiconductors orglass, forming fluid flow microchannels in silicon, and in general forproviding a variety of mechanical features in a substrate. In measuringstripe widths and separations the prior art techniques have typicallyrelied on scanning electron microscopy (SEM). Unfortunately, SEM is adestructive and very time-consuming examination method.

[0007] Methods such as atomic force microscopy (AFM) and profilometryare also viable for examining features or patterns of features. However,both of these methods are very time consuming and they require specialtest structures in most cases.

[0008] The patterning masks used to create resist lines often themselvescontain features. Of particular interest are Alternating Phase ShiftMasks (AAPSMs), which are often quartz or fused silica plates etchedwith trenches in repeating patterns. This creates an interferencecondition between light passing through the etched and un-etched regionsof the mask, leading to complete amplitude cancellation in regions thatwould normally have been exposed. In this way an AAPSM can be used topattern features in the resist that are smaller than the wavelength oflight used to expose the resist. Accurate metrology control of thedimensions of these features is critical, since in a typical applicationusing 248 nm wavelength light, approximately 13 Å difference in trenchdepth is enough to change the phase shift by 1 degree.

[0009] In addition to AFM, SEM, and profilometry the prior art offersinterferometric techniques for measuring high-precision patterns, suchas those encountered in AAPSMs. Unfortunately, because of the inherentlimitations of AFM, SEM and profilometry already mentioned, thesetechniques are not satisfactory for examining AAPSMs. Interferometrictechniques are too expensive, and require special test structures.Furthermore, the test features have to be large enough so that referenceand measurement beams can be fully covered by two different uniformareas respectively. These test features often do not reflect the phaseshift characteristics of the features to be printed on the mask. Inaddition, in most cases the test features have to be transparent. Thiscondition prevents the measurement from being performed at the earlystages of mask processing when an opaque metallic layer, such as Cr, isfrequently present.

[0010] For more information on AAPSMs and methods for examining them thereader is referred to Cynthia B. Brooks, et al., “Process Monitoring ofEtched Fused Silica Phase Shift Reticles”, Proceedings of the SPIE,22^(nd) Annual BACUS Symposium on Photomask Technology and Management,Sep. 30-Oct. 4, 2002, Monterey, Calif., USA; Alessandro Callegari andKatherina Babich, “Optical Characterization of Attenuated PhaseShifters”, SPIE, Vol. 3050, pp. 507-514; as well as Pieter Burggraaf,“Lithography's Leading Edge, Part I: Phase Shift Technology”, February1992, pp. 43-47.

[0011] More recently, attempts have been made to measure patterns usingscatterometry. In this technique, a pattern is subjected to light, suchas from a laser, typically having a single wavelength. This light isusually directed toward the pattern at some angle to the normal. Thelight reflected from the pattern at various diffracted orders ismeasured. It may be possible to use such data to obtain quantitativeinformation about the pattern. However, scatterometry is very sensitiveto small changes in the profile of the pattern, and requires relativelysophisticated correlation work to relate the reflected radiation to thefeatures of a pattern. The computational effort required to correlatethe reflected radiation to the pattern is very high since theconvergence criteria for these solutions take a very long time tocompute. In addition, the measured pattern must be periodic. Otherexamples of characterization methods pertaining to photolithography andequipment suitable for practicing such methods are described in U.S.Pat. Nos. 5,867,276; 5,363,171; 5,184,021; 4,866,782 and 4,757,207.There are still other types of scatterometry, which measure thespecularly reflected light as a function of wavelength, as taught inU.S. Pat. Nos. 6,483,580; 5,963,329; 5,739,909 and 5,607,800.

[0012] Of these references U.S. Pat. No. 5,607,800 to Ziger teaches amethod and arrangement for characterizing features of a patternedmaterial on an underlayer. His approach is based on selecting anappropriate wavelength range where the patterned material absorbs moreradiation than the underlayer. In other words, substrate or underlayeris more reflective than the pattern or surface features in thiswavelength range. The reflectance spectrum uniquely identifies thepattern and can be used to study similar patterns by comparing theirreflectance spectra. Unfortunately, just as in the case ofscatterometry, when patterns vary this comparison-based approach can notbe used effectively to study patterns which differ substantially fromeach other.

[0013] U.S. Pat. No. 6,100,985 to Scheiner et al. teaches a method formeasuring at least one desired parameter of a patterned structure havinga plurality of features. In this method a measurement area, which issubstantially larger than a surface area of the structure defined by thegrid cycle, is illuminated by an incident radiation of a presetsubstantially wide wavelength range. The light component that issubstantially specularly reflected from the measurement area isdetected, and measured data representative of photometric intensities ofeach wavelength within the wavelength range is obtained. The measuredand theoretical data are analyzed and the optical model is optimizeduntil the theoretical data satisfies a predetermined condition. Upondetecting that the predetermined condition is satisfied at least oneparameter of the structure can be calculated.

[0014] A still more recent teaching for optically determining a physicalparameter of a pattern made up of features is taught in U.S. Pat. No.6,327,035 to Li et al. This teaching goes further than Scheiner et al.by examining various response light fractions including an underlayerlight fraction and a feature light fraction and using reference physicalparameters of the underlayer. The response light can be eithertransmitted or reflected and the reference physical parameters of theunderlayer are either known a priori or determined.

[0015] U.S. Pat. No. 6,340,602 to Johnson et al. teaches a method formeasuring a parameter associated with a portion of a sample having oneor more structures with at least two zones each having an associatedzone reflectance property. The at least two zones are illuminated withbroadband light, the reflected light is measured and a measuredreflectance property is fit to a model. The model mixes the zonereflectance properties to account for partially coherent lightinteractions between the two zones.

[0016] Although Johnson's approach attempts to address coherence issuesbetween the zones, it does not take into account the interactionsbetween the broadband light and the substrate. More precisely, in thisapproach the substrate is assumed to be opaque and only lateralincoherence between the zones themselves is treated. In most cases,however, substrates on which features or zones are measured are at leastpartially transparent over a portion or even the entire broadbandspectrum of the incident broadband light. Thus, by leaving out thecomplex interactions between the illuminating light, the zones and thesubstrate, Johnson is not able to provide a method that can be used formeasuring zones or features on semi-transparent and transparentsubstrates.

[0017] In fact, the problem of optically examining features and patternson underlayers or substrates that are at least semi-transparent or fullytransparent has eluded a satisfactory solution because of itscomplexity. This complexity is partly due to the large series ofinternal reflections and transmissions affecting the response lightobtained from the substrate and features. What is more, the responselight is not only conditioned by the multiple internal reflections andtransmissions within the substrate and features to be examined, but alsoby coherent and incoherent interactions between reflected and/ortransmitted response light from the substrate and the various features.

OBJECTS AND ADVANTAGES

[0018] In view of the above, it is a primary object of the presentinvention to provide a method and apparatus that enables a thoroughoptical examination of features on semi-transparent and even transparentsubstrates. More specifically, it is an object of the present inventionto provide a method of examining the response light in a manner whichtakes into account the coherent and incoherent interactions betweenreflected and/or transmitted response light within and among variousfeatures.

[0019] These and numerous other objects and advantages of the presentinvention will become apparent upon reading the following description.

SUMMARY

[0020] The objects and advantages of the present invention are securedby a method for determining a physical parameter of features on asubstrate by illuminating the substrate with an incident light coveringan incident wavelength range Δλ where the substrate is at leastsemi-transparent and such that the incident light enters the substrateand the features. A response light received from the substrate and thefeatures is measured to obtain a response spectrum of the responselight. Further, a complex-valued response due to the features and thesubstrate is computed and both the response spectrum and thecomplex-valued response are used in determining the physical parameter.This physical parameter can be a depth, a width, a real part of thecomplex refractive index, an imaginary part of the complex refractiveindex or some other physical parameter of the features.

[0021] The response light is reflected light, transmitted light or acombination of the two and it can be either polarized or unpolarized.Thus, the response spectrum corresponds to either a reflectance R, atransmittance T or both. The complex-valued response typically includesa complex reflectance amplitude, a complex transmittance amplitude orboth. In accordance with the method of invention, when thecomplex-valued response is or includes the complex reflectance amplitudethe reflectance R is computed by multiplying the complex reflectanceamplitude with its complex conjugate. Similarly, when the complex-valuedresponse is or includes the complex transmittance amplitude thetransmittance T is computed by multiplying the complex transmittanceamplitude with its complex conjugate.

[0022] A vertical coherence length L_(vc) of the incident light andthickness d_(s) of the substrate determine whether the response light iscoherent or incoherent. For example, when the vertical coherence lengthL_(vc) is small with respect to thickness d_(s) then the response lightexhibits incoherence. In such cases a phase δ_(s) of the complex-valuedresponse is averaged in the computation.

[0023] The method of invention is particularly advantageous when thefeatures are adjacent. In most such cases the wavelength range Δλ isselected such that the substrate and the adjacent features produce acoherent fraction and an incoherent fraction in the response light.Preferably, further computations are made to determine a coherentfraction β (or coherent factor) for coherent adding of thecomplex-valued response. An incoherent fraction of the response light isequal to 1−β. The coherent fraction β can be determined from a lateralcoherence length L_(lc) of the incident light. It should be noted thatthis approach presents a closed-form solution to determining thecomplex-valued response of adjacent features on a substrate.

[0024] In cases where the features are periodic the incident light willexperience diffraction. Thus, when the features are periodic it ispreferable to focus the incident light to an illumination area coveringa sufficiently small number of features such that diffraction effectsare negligible.

[0025] When the area of the features is larger than the lateralcoherence length L_(lc) of the incident light then the complex-valuedresponse from the features is added incoherently. Otherwise, when thearea of at least one of the features is smaller than the lateralcoherence length L_(lc) then the complex-valued response is addedcoherently. It should be noted that lateral coherence length L_(lc) aswell as vertical coherence length L_(vc) are wavelength dependent.

[0026] The features can be adjacent features made of two differentmaterials, such as material 1 covering a first area fraction a₁ andmaterial 2 covering a second area fraction a₂. The area fractions a₁ anda₂ correspond to the fractional area illuminated by the incident light.Depending on the embodiment, the complex-valued response to be addedcoherently is a total complex-valued reflectance amplitude r_(C), atotal complex-valued transmittance amplitude t_(C) or both. Thefollowing equations are used in the computations:

r _(C) =a ₁ r ₁ +a ₂ r ₂,

t _(C) =a ₁ t ₁ +a ₂ t ², and

a ₁ +a ₂=1.

[0027] The response spectra such as a coherent reflectance R_(C) andcoherent transmittance T_(C) are then computed by multiplying out thecomplex-valued amplitudes by their complex conjugates. In particular,coherent reflectance R_(C) is computed by using the following crossterm:${{\langle{r_{1} \cdot r_{2}^{*}}\rangle} = \frac{{r_{1,{as}}r_{2,{as}}^{*}} + {\left( {{t_{1,{as}}t_{2,{as}}^{*}t_{1,{sa}}t_{2,{sa}}^{*}} - {r_{1,{as}}r_{2,{as}}^{*}r_{2,{sa}}r_{2,{sa}}^{*}}} \right)r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}}{1 - {r_{1,{sa}}r_{2,{sa}}^{*}r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}}},$

[0028] where α_(s) is an absorption coefficient of the substrate andd_(s) is the thickness of the substrate. In embodiments where theincident light is focused on a back side of the substrate, the crossterm simplifies and is computed as:

r ₁ ·r ₂

*

=t _(1,as) t _(2,as) *t _(1,sa) t _(2,sa) *r _(1,sb) r _(2,sb) *e ^(−2α)^(_(s)) ^(d) ^(_(s.))

[0029] Meanwhile, coherent transmittance T_(C) is calculated by usingthe following cross term:${{\langle{t_{1} \cdot t_{2}^{*}}\rangle} = {\frac{t_{1,{as}}t_{2,{as}}^{*}t_{1,{sb}}t_{2,{sb}}^{*}^{{- \alpha_{s}}d_{s}}}{1 - {r_{1,{sa}}r_{2,{sa}}^{*}r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}} = {A\quad ^{\varphi}}}},$

[0030] where A is the amplitude of

t₁·t₂*

, φ is the phase shift between t₁ and t₂. As before, α_(s) is theabsorption coefficient and d_(s) is the thickness of the substrate.

[0031] The method of invention can be practiced under variousillumination conditions. In one embodiment the incident light iscollimated. In another embodiment, the incident light is focused. Forexample, the incident light is focused on a surface of the substrate.The substrate can be illuminated from a first side where the feature orfeatures are located or from a side opposite the first side. It shouldalso be noted that the incident light can be linearly polarized.

[0032] In certain applications of the method the physical parameter isderived from phase shift φ or amplitude A of the response light. Inother words, phase shift φ and/or variation of amplitude A experiencedby reflected and/or transmitted response light is used to determine thephysical parameter of the features. In some embodiments, informationabout physical parameters of the features is derived from phase shift φ.Specifically, in embodiments where the response light is transmitted thephase shift φ can be obtained from:$\varphi = {\varphi_{T} = {\frac{2{\pi \left( {{n\quad \cos \quad \theta_{2}} - {\cos \quad \theta_{1}}} \right)}t_{s}}{\lambda}.}}$

[0033] In embodiments where the response light is reflected the phaseshift φ can be obtained from:$\varphi = {\varphi_{R} = {\frac{4\pi \quad n\quad t_{s}\cos \quad \theta_{2}}{2}.}}$

[0034] In the preferred embodiment where at least two adjacent featuresare being examined the incident light enters the substrate and thefeatures and the complex-valued response exhibits interference due tothe features. The interference manifests in phase φ observed in themeasured response light. In order to examine these variations it isconvenient to examine a wide reflectance R and/or transmittance Tspectrum, e.g., from about 190 nm to about 1000 nm.

[0035] The method of invention can be used to determine physicalparameters of features in various arrangements. For example, at leastone of the features can be in the form of a film, e.g., a flat film. Oneor more additional features can be embedded in the film.

[0036] The invention further extends to an apparatus for determining aphysical parameter of one or more features on a substrate. The apparatushas an illumination source for producing the incident light and opticsfor guiding the incident light such that the incident light enters thesubstrate and the features. A detector is provided for receiving theresponse light and measuring its response spectrum. In addition, theapparatus has a processing unit for computing the complex-valuedresponse of the substrate and the one or more features and determiningthe physical parameter from the measured response spectrum and thecomplex-valued response.

[0037] The substrate can be transparent within the incident wavelengthrange Δλ or optically thick. In practice, the substrate will exhibit avariation in its degree of transparency over the selected wavelengthrange Δλ. In some cases, the substrate can be optically thick over alarge portion of the entire wavelength range Δλ, e.g., when thesubstrate comprises a metal layer. In a preferred embodiment theapparatus examines a wide wavelength range Δλ by providing anillumination source that is broadband. In one embodiment the broadbandillumination source provides an incident wavelength range Δλ from about190 nm to about 1000 nm.

[0038] A detailed description of the invention and the preferred andalternative embodiments is presented below in reference to the attacheddrawing figures.

BRIEF DESCRIPTION OF THE FIGURES

[0039]FIG. 1 (PRIOR ART) is a schematic diagram illustrating some opticsprinciples on which the invention is based.

[0040]FIG. 2 is a diagram illustrating an apparatus according to theinvention for examining a semi-transparent substrate with adjacentfeatures.

[0041]FIG. 3 illustrates a cross-sectional view of a fused silica sampleetched with adjacent features.

[0042]FIG. 4 are graphs of reflectance and transmittance spectra R, Tfor the sample of FIG. 3.

[0043]FIG. 5 is a graph of phase shift for the sample of FIG. 3.

[0044]FIG. 6 are graphs of transmittance T spectra for samples analogousto that of FIG. 3 and having varying area fractions of trenches.

[0045]FIG. 7 are graphs of normalized reflectance spectra from back side(b) for samples analogous to that of FIG. 3 and having varying areafractions of trenches.

[0046]FIG. 8 are graphs of normalized reflectance spectra from frontside (a) for samples analogous to that of FIG. 3, normalized relative toa uniform sample, and having varying area fractions of trenches.

[0047]FIG. 9 is a cross sectional view of a portion of another fusedsilica sample examined with the method of the invention.

[0048]FIG. 10 is a cross sectional view of a portion of yet anotherfused silica sample examined with the method of the invention.

[0049]FIG. 11 are graphs of reflectance R spectra measured on the sampleof FIG. 10 from the front side and the back side.

[0050]FIG. 12 (PRIOR ART) is a diagram of a standard transmission-typeinterferometric apparatus examining a substrate with features.

[0051]FIG. 13 is a diagram illustrating the method of invention usingresponse light reflected from the front side of the substrate withfeatures as shown in FIG. 12.

[0052]FIG. 14 is a diagram illustrating the method of invention usingresponse light reflected from the back side of the substrate withfeatures as shown in FIG. 12.

[0053]FIG. 15 is a diagram illustrating the method of invention usingtransmitted response light from substrate with features as shown in FIG.12.

[0054]FIG. 16 illustrates a portion of another apparatus according tothe invention.

THEORETICAL BACKGROUND

[0055] The instant invention will be best understood by firstconsidering the prior art schematic diagram of FIG. 1 illustrating someoptics principles. Apparatus 10 has an illumination source 12 thatgenerates an incident light 14 spanning an incident wavelength range Δλ.Source 12 is positioned to illuminate a substrate 16 with incident light14. Optics 18 are positioned to guide incident light 14 from source 12to substrate 16.

[0056] Substrate 16 has a thickness d_(s) typically on the order of afraction of a millimeter to several millimeters, e.g., 0.2 mm to 8 mm.Substrate 16 can be made of any material that is semi-transparent withinan incident wavelength range Δλ covered by incident light 14. Thematerial of substrate 16 is optically described by a real part n_(s) andimaginary part k_(s) of the complex refractive index. For example, thematerial of substrate 16 can be diffused silica or glass.

[0057] Substrate 16 has a stack of films 36 on a first side a and astack of films 38 on a second side b. Stack 36 can include a largenumber of films 36A, 36B, . . . 36N. Likewise, stack 38 can include alarge number of films 38A, 38B, . . . 38N.

[0058] Incident light 14 enters substrate 16 through stack 36 depositedon a side a of substrate 16. It should be noted that films 36 can haveany structure and composition and represent features. Now, when incidentlight 14 enters substrate 16 from side a two complex-valued responsesdue to substrate 16 and stacks 36, 38 are produced. The firstcomplex-valued response is a complex reflectance amplitude r and thesecond complex-valued response is a complex transmittance amplitude t,given by: $\begin{matrix}{{r = \frac{r_{as} + {\left( {{t_{as}t_{sa}} - {r_{as}r_{sa}}} \right)r_{sb}^{{- {\delta}_{s}} - {\alpha_{s}d_{s}}}}}{1 - {r_{sa}r_{sb}^{{- {\delta}_{s}} - {\alpha_{s}d_{s}}}}}},} & {{Eq}.\quad 1} \\{t = {\frac{t_{as}t_{sb}^{{- {({{\delta}_{s} + {\alpha_{s}d_{s}}})}}/2}}{1 - {r_{sa}r_{sb}^{{- {\delta}_{s}} - {\alpha_{s}d_{s}}}}}.}} & {{Eq}.\quad 2}\end{matrix}$

[0059] In these equations δ_(s) is the phase and α_(s) is the absorptioncoefficient given by: $\begin{matrix}{{\delta_{s} = \frac{4\pi \quad n_{s}d_{s}\cos \quad \theta_{s}}{\lambda}},} & {{Eq}.\quad 3} \\{\alpha_{s} = {\frac{4\pi \quad k_{s}\cos \quad \theta_{s}}{\lambda}.}} & {{Eq}.\quad 4}\end{matrix}$

[0060] In these equations “a” denotes side a, “b” denotes side b, sdenotes substrate 16, d₅ is the thickness of substrate 16, n_(s), k_(s)are the real and imaginary parts of the complex refractive index ofsubstrate 16 and θ_(s) is the incident angle of light 14 insidesubstrate 16. Coefficients r_(uv) and t_(uv) (u,v=a,b,s) are thereflection and transmission coefficients. For example, t_(as) is thetransmission coefficient from the atmosphere surrounding substrate 16,in this case air, through stack 36 on side a to substrate 16, and t_(sa)is the transmission coefficient from substrate 16 through stack 36 onside a to air. The analytical expressions for r_(uv), and t_(uv) arewell known and can be found in standard textbooks, such as O. S.Heavens, Optical Properties of Thin Solid Films, Dover, Chapter 4. Itshould be noted that these equations ate valid for both s- andp-polarized light.

[0061] In response to incident light 14 substrate 16 and stacks of films36, 38 generate response light. The response light includes bothreflected light 24 and transmitted light 28. The response spectrum ofreflected light 24 detected by detector 26 is obtained by multiplyingcomplex reflectance amplitude r by its complex conjugate r*. Thismultiplication yields a reflectance R:

R=r·r*.  Eq. 5

[0062] Similarly, the response spectrum of transmitted light 28 detectedby detector 30 is described by a transmittance T. Transmittance T isobtained by multiplying complex transmittance amplitude t by its complexconjugate t* as follows:

T=t·t*.  Eq. 6

[0063] The choice of range Δλ of incident light 14 is such thatsubstrate 16 is semi-transparent or event transparent at any particularwavelength, e.g., at λ_(i), within range Δλ. Therefore, at a particularwavelength, e.g., at λ_(i), response light 24, 28 undergoes multipleinternal reflections and transmissions before emerging from substrate16.

[0064] Depending on a vertical coherence length L_(vc) of light 14response light 24, 28 is coherent or incoherent. More precisely, whenvertical coherence length L_(vc) is sufficiently small with respect tothickness d_(s) of substrate 16 then response light 24, 28 exhibitsincoherence. This is visualized in FIG. 1 for response light 24, 28 atwavelength λ_(i) by showing a “slip-off” in phase δ_(s) produced afterseveral internal reflections. For a source 12 with 2 nm line width(which gives vertical coherence length L_(vc)˜0.1 mm at 500 nm) andthickness d_(s) larger than 0.2 mm response light 24, 28 undergoesmultiple reflections within substrate 16 and exhibits incoherence.

[0065] When response light 24, 28 exhibits incoherence then phase δ_(s)of the complex-valued responses, i.e., complex reflectance andtransmission amplitudes r, t needs to be averaged. Averaging of phaseδ_(s) yields the following reflectance R and transmittance T:$\begin{matrix}\begin{matrix}{{R = {\frac{1}{2\pi}{\int_{0}^{2\pi}{{r \cdot r}*{\delta_{s}}}}}},{or}} \\{{= \frac{{r_{as}r_{as}^{*}} + {\left( {{t_{as}t_{as}^{*}t_{sa}t_{sa}^{*}} - {r_{as}r_{as}^{*}r_{sa}r_{sa}^{*}}} \right)r_{sb}r_{sb}^{*}^{{- 2}\alpha_{s}d_{s}}}}{1 - {r_{sa}r_{sa}^{*}r_{sb}r_{sb}^{*}^{{- 2}\alpha_{s}d_{s}}}}},}\end{matrix} & {{Eq}.\quad 7} \\\begin{matrix}{{T = {\frac{1}{2\pi}{\int_{0}^{2\pi}{{t \cdot t}*{\delta_{s}}}}}},{or}} \\{= {\frac{{t_{as}t_{as}^{*}t_{sb}t_{sb}^{*}^{{- \alpha_{s}}d_{s}}}}{1 - {r_{sa}r_{sa}^{*}r_{sb}r_{sb}^{*}^{{- 2}\alpha_{s}d_{s}}}}.}}\end{matrix} & {{Eq}.\quad 8}\end{matrix}$

[0066] The response spectra, in this case reflectance R andtransmittance T, account for the complex-valued transmittance andreflectance amplitudes r, t due to substrate 16 and stacks 36, 38.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0067] The method of invention is based on the fact that knowledge ofmaterials making up films 36, 38 and substrate 16, as well as theirphysical dimensions yield the information necessary to computecomplex-valued reflectance and transmittance amplitudes r, t. Therefore,in practicing the method of invention the complex-valued response due tothe features such as films 36, 38 and substrate 16 has to be knownbefore measurement. The computed complex-valued response is used indetermining one or more physical parameters of one or more of films 36,38 actually being examined on substrate 16.

[0068] The method of invention finds its preferred application inmeasuring physical properties of features that are adjacent to eachother and are located on a semi-transparent or even transparentsubstrate. FIG. 2 illustrates an apparatus 50 in accordance with theinvention for determining one or more physical parameters of features52A, 52B, . . . 52N on a uniform substrate 54. Features 52A, 52B, . . .52N are made of two different materials and are positioned adjacent eachother. In this embodiment odd numbered features, i.e., 52A, 52C, 52E, .. . etc. are made of material 1 such as amorphous fused silica and evennumbered features, i.e., 52B, 52D, 52F, . . . etc. are made of material2 such as air (air gaps). This type of arrangement of features 52 isencountered, for example, in an Alternating Aperture Phase Shift Mask(AAPSM). It should be understood, that the arrangement of features 52can be periodic or non-periodic and that features 52 can be made of morethan two types of materials.

[0069] In addition to adjacent features 52, substrate 54 also carries anumber of features in the form of layers 53A, 53B and 53C. Layers 53 canbe made of materials different from those of features 52.

[0070] Apparatus 50 has an illumination source 56. Illumination source56 generates an incident light 58 spanning an incident wavelength rangeΔλ. Preferably source 56 is broadband and its range Δλ extends fromabout 190 nm to about 1000 nm. To cover such range source 56 can includea number of individual sources spanning separate or even overlappingportions of wavelength range Δλ. Substrate 54 is at leastsemi-transparent within wavelength range Δλ. Of course, the actual levelof transparency of substrate 54 to incident light 58 may differ greatlybetween different wavelengths within range Δλ.

[0071] Apparatus 50 is equipped with an optic 60 for guiding incidentlight 58 from source 56 to substrate 54. Optic 60 is shown in the formof a lens, but it is understood that a compound optic system can be usedas optic 60. In particular, it is preferred that optic 60 have a beamshaping power such that it can focus or collimate incident light 58 onsubstrate 54, as indicated in dashed and dotted lines.

[0072] An optic 62 is positioned above substrate 54 to guide responselight 64 generated in response to incident light 58 by substrate 54 andfeatures 52, 53 to a detector 66. Although optic 62 is shown in the formof a lens, it is understood that a compound optic system can be used asoptic 62. Response light 64 is a reflected light and it has a responsespectrum corresponding to a reflectance R of substrate 54 and features52, 53. An additional optic 68 and detector 70 can be provided to alsocollect a response light 72 transmitted through features 52, 53 andsubstrate 54. Response light 72 has a response spectrum corresponding toa transmittance T of features 52, 53 and substrate 54.

[0073] Apparatus 50 has a detector 66 for receiving response light 64and a detector 70 for receiving response light 72. Detectors 66, 70 areany suitable photodetectors for receiving response light 64, 72respectively and measuring a response spectrum of response light 64, 72.More specifically, detectors 66, 70 are designed to measure responselight 64, 72 over a response spectrum covering entire wavelength rangeΔλ.

[0074] Furthermore, each detector 64, 70 is connected to a processingunit 74, 76 respectively. Processing units 74, 76 analyze the responsespectra and perform computations to obtain computed reflectance andtransmittance amplitudes r, t or else import reflectance andtransmittance amplitudes r, t from elsewhere. In the embodiment shownunits 74, 76 can be in communication with each other, as indicated bythe dashed line. It should be noted that in alternative embodimentsunits 74, 76 can be combined in a single processing device or unit.

[0075] During determination of one or more physical parameters offeatures 52, 53 detectors 66, 70 receive response light 64, 72 fromsubstrate 54 and features 52, 53. Detectors 66, 70 measure the responsespectrum of response light 64, 72. In this case detector 66 measuresreflectance R and detector 70 measures transmittance T over wavelengthrange Δλ. Deviations of reflectance R and transmittance T measured bydetectors 66, 70 from the values computed using the complex-valuedreflectance and transmittance amplitudes r, t are used by processingunits 74, 76 to determine the one or more physical parameters of one ormore of features 52, 53. The physical parameter or parameters caninclude a depth or thickness, a width, a real part of the complexrefractive index, an imaginary part of the complex refractive index orsome other physical parameter of any one of features 52, 53.

[0076] Processing units 74, 76 determine the one or more physicalproperties of features 52, 53 based on a computed complex-valuedresponse for a sample of desired dimensions and material composition andmeasured response spectrum or spectra, such as the reflectance R and/ortransmittance T. In some embodiments the complex-valued response can becomputed with the aid of measurements of a suitable reference samplethat is built of substrate 54 with features 52, 53 just like the samplesto be tested. Alternatively, the complex-valued response can be derivedpurely mathematically.

[0077] During operation, optic 60 guides incident light 58 such that itis incident on an area composing a number of features 52, specificallyfeatures 52A through 52N. Since in the present embodiment features 52may form a periodic structure the area illuminated by incident light 58should be kept small to avoid possible diffraction effects. Preferably,the area illuminated by incident light 58 should cover a sufficientlysmall number of features 52 such that diffraction effects arenegligible. For example, when working in wavelength range Δλ from 190 nmto 1000 nm with features 52 on the order of tens to hundreds ofnanometers, the number of features 52 illuminated by incident light 58should be kept under 50 and even under 20 in some cases. This conditionis not as crucial or may even be unnecessary when features 52 are notarranged in a periodic structure. To take into account the diffractioneffects the complex response has to be computed using vector theory.Additional information and relevant teaching on the application of thevector theory can be found in U.S. Pat. Nos. 6,483,580; 5,963,329;5,739,909 and 5,607,800.

[0078] Incident light 58 has a lateral coherence length L_(lc) asindicated. It should be noted that, in general, lateral coherence lengthL_(lc) is not equal and may differ quite substantially from verticalcoherence length L_(vc). When the size of adjacent features 52 issmaller than coherence length L_(lc) the response light 64, 72 fromadjacent features 52 is added coherently. Thus, when the area fractionscovered by features 52 made of material 1 (i.e., features 52A, 52C, . .. ) and made of material 2 (i.e., features 52B, 52D, . . . ) correspondto a₁ and a₂, respectively, then the total complex-valued reflectanceand transmittance amplitudes are given by the combination of amplitudesof response light 64, 72 from these two areas, as follows:

r _(C) =a ₁ r ₁ +a ₂ r ₂,  Eq. 9

t _(C) =a ₁ t ₁ +a ₂ t ₂,  Eq. 10

a ₁ +a ₂=1,  Eq. 11

[0079] where the subscript “C” denotes coherent adding. It should benoted that r₁, r₂, t₁, t₂ are the complex-valued reflectance andtransmittance amplitudes for areas 1 and 2, respectively, and they maybe calculated in accordance with equations 1-2 as discussed above. Theresponse spectra such as a coherent reflectance R_(C) and a coherenttransmittance T_(C) are computed by multiplying out the complex-valuedreflectance and the complex-valued transmittance by their respectivecomplex conjugates as follows: $\begin{matrix}\begin{matrix}{R_{C} = {\left( {{a_{1}r_{1}} + {a_{2}r_{2}}} \right) \cdot \left( {{a_{1}r_{1}} + {a_{2}r_{2}}} \right)^{*}}} \\{= {{a_{1}^{2}R_{1}} + {a_{2}^{2}R_{2}} + {2a_{1}{a_{2} \cdot {{Real}\left( {r_{1} \cdot r_{2}^{*}} \right)}}}}}\end{matrix} & {{Eq}.\quad 12} \\\begin{matrix}{T_{C} = {\left( {{a_{1}t_{1}} + {a_{2}t_{2}}} \right) \cdot \left( {{a_{1}t_{1}} + {a_{2}t_{2}}} \right)^{*}}} \\{= {{a_{1}^{2}T_{1}} + {a_{2}^{2}T_{2}} + {2a_{1}{a_{2} \cdot {{Real}\left( {t_{1} \cdot t_{2}^{*}} \right)}}}}}\end{matrix} & {{Eq}.\quad 13}\end{matrix}$

[0080] where R₁=(r₁·r₁*), R₂=(r₂·r₂*), T₁=(t₁·t₁*), and T₂=(t₂·t₂*) arereflectances and transmittances from areas 1 and 2 respectively.

[0081] As discussed above, when substrate 54 is thick, phase δ_(s) hasto be averaged. R₁, R₂, T₁, and T₂ can then be calculated from equations7 and 8. The cross terms are given by: $\begin{matrix}\begin{matrix}{\left. {r_{1} \cdot r_{2}^{*}}\Rightarrow{\langle{r_{1} \cdot r_{2}^{*}}\rangle} \right. = {\frac{1}{2\pi}{\int_{0}^{2\pi}{{r_{1} \cdot r_{2}^{*}}{\delta_{s}}}}}} \\{= \frac{{r_{1,{as}}r_{2,{as}}^{*}} + {\left( {{t_{1,{as}}t_{2,{as}}^{*}t_{1,{sa}}t_{2,{sa}}^{*}} - {r_{1,{as}}r_{2,{as}}^{*}r_{2,{sa}}r_{2,{sa}}^{*}}} \right)r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}}{1 - {r_{1,{sa}}r_{2,{sa}}^{*}r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}}} \\{\left. {t_{1} \cdot t_{2}^{*}}\Rightarrow{\langle{t_{1} \cdot t_{2}^{*}}\rangle} \right. = {\frac{1}{2\pi}\frac{1}{2\pi}{\int_{0}^{2\pi}{{t_{1} \cdot t_{2}^{*}}{\delta_{s}}}}}} \\{= \frac{{t_{1,{as}}t_{2,{as}}^{*}t_{1,{sb}}t_{2,{sb}}^{*}^{{- \alpha_{s}}d_{s}}}}{1 - {r_{1,{sa}}r_{2,{sa}}^{*}r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}}} \\{= {A\quad ^{\varphi}}}\end{matrix} & {{Eq}.\quad 14}\end{matrix}$

[0082] In these equations subscripts 1 and 2 represent areas 1 and 2,respectively. A is the amplitude of <t₁·t₂*> and φ is the phasedifference or shift between t₁ and t₂. A person skilled in the art willrecognize that equation 15 provides a very convenient way for measuringand calculating the phase shift for phase masks such as AAPSMs.Equations 12-15 are novel in providing an analytic and closed formexpression for coherent lateral interference between adjacent featureson thick transparent or semi-transparent substrates. It should be notedthat knowledge of phase shift φ will be sufficient in some cases to makesome determination about physical parameters of features 52, and is avaluable piece of information in and of itself, as will be appreciatedby those skilled in the art.

[0083] Light 24, 28 have response spectra which are influenced bycomplex reflectance and transmittance amplitudes r_(C) and t_(C).Specifically, the presence of the cross terms <r₁·r₂*> and <t₁·t₂*> inequations 12 and 13 affects the response spectra such as the reflectanceR and transmittance T over wavelength range Δλ. In fact, because of thecross terms the total reflectance R and total transmittance T withinrange Δλ experience interference effects such their sum may be less than1 (assuming no absorption losses).

[0084] Equations 14-15 can be simplified in some cases. For example,when substrate 54 is thick and highly absorbing such that:

α_(s)d>>1 or e^(−α) ^(_(s)) ^(d) ^(_(s)) ≈0.

[0085] then the expressions for the cross terms can be simplified asfollows:

r ₁ ·r ₂

*

=r _(1,as) r _(2,as)*, and  Eq. 16

t ₁ ·t ₂*

=0.  Eq. 17

[0086] For measurement purposes, it is sometimes convenient for optic 60to produce a focused beam of incident light 58 rather than a collimatedbeam. One extreme case is when the depth of field is much shorter thanthickness d_(s) of substrate 54. If the beam of light 58 is focused onone side of substrate 54, e.g., on side a, then the reflectance from theother side, i.e., side b of substrate 54 will not be detected.Therefore, when the beam of light 58 is focused on side a, which is thefront side or front surface, equation 14 will be changed to equation 16by setting r_(i,sb)=0. On the other hand, when the beam of light 58 isfocused on side b, which is the back side or back surface, equation 14will be changed (r_(i,sa)=0, r_(i,as)=0) to:

r ₁ ·r ₂

*

=t _(1,as) t _(2,as) *t _(1,sa) t _(2,sa) *r _(1,sb) r _(2,sb) *e ^(−2α)^(_(s)) ^(d) ^(_(s)) .  Eq. 18

[0087] R₁ and R₂ in equation 12 need to be modified accordingly.

[0088] When the size of area 1 and 2 is much larger than lateralcoherence length L_(lc) of incident light 58, then response light 64, 72from those two areas add incoherently. Thus, the total reflectance andtransmittance are given by:

R ₁ =a ₁ R ₁ +a ₂ R ₂,  Eq. 19

T ₁ =a ₁ T ₁ +a ₂ T ₂,  Eq. 20

[0089] where the subscript “I” denotes incoherent adding. It should benoted that equations 9-10 and 19-20 can be extended to cases wherethree, four or even more different areas are illuminated.

[0090] In most practical embodiments, response light 64, 72 from area 1and area 2 are partially coherent. Thus, the reflectance R andtransmittance T including the contributions of both coherent andincoherent fractions to their response spectra can be described asfollows:

R=(1−β)R ₁ +βR _(C),  Eq. 21

T=(1−β)T ₁ +βT _(C),  and Eq. 22

0≦β≦1  Eq. 23

[0091] where β is a fraction for coherent adding. Now, when material 1and material 2 are identical through the whole stack (i.e., substrate 54and features 52, 53) then R=R_(I)=R_(C)=R₁=R₂ and T=T_(I)=T_(C)=T₁=T₂independent of b, a₁ and a₂.

[0092] Fraction β, also called the coherence fraction, is related tolateral coherence length L_(lc) as follows: $\begin{matrix}{L_{lc} = \frac{\lambda^{2}}{{\Delta\lambda}_{{spect}.}}} & {{Eq}.\quad 24}\end{matrix}$

[0093] where λ is the wavelength and Δλ_(spect.) is the spectral bandwidth of detectors 66, 70 and preferably covers the entire bandwidth Δλof incident light 58. For more information the reader is referred toGrant R. Fowles, Introduction to Modern Optics, Second Edition, Dover,1975, p. 73. Usually, the first order of diffracted response light isused and the grating equation is given by:

λ=p(sin θ_(i)+sin θ_(r))  Eq. 25

[0094] where p is the period of the grating, and θ_(i), θ_(r) are theangles for incident and diffracted light. This equation can bere-written as:

Δλ={square root}{square root over (p ²−(λ−p sin θ_(i))²)}Δθ_(r),  Eq. 26

[0095] where Δθ_(r) is the angular spread of the diffracted responselight. Using equation 26 coherent fraction β can be approximated by:$\begin{matrix}\begin{matrix}{{\beta = {{\frac{\beta_{1}\lambda^{2}}{\beta_{0}\sqrt{p^{2} - \left( {\lambda - {p\quad \sin \quad \theta_{i}}} \right)^{2}}}\quad {when}\quad \beta} < 1}},{and}} \\{\beta = {1\quad {{otherwise}.}}}\end{matrix} & {{Eq}.\quad 27}\end{matrix}$

[0096] In equation 27 β₁ is the coherent factor (wavelength independent)and β_(o) is the normalization factor given by: $\begin{matrix}{{\beta_{0} = \frac{\lambda_{o}^{2}}{\sqrt{p^{2} - \left( {\lambda_{o} - {p\quad \sin \quad \theta_{i}}} \right)^{2}}}},} & {{Eq}.\quad 28}\end{matrix}$

[0097] where λ_(o) is the shortest wavelength in the collected spectrumΔλ_(i).

[0098] The above equations are used to determine the response spectra,i.e., reflectance R and transmittance, T that should be observed bydetectors 66, 70 when the sample being measured conforms to therequirements. In practice, processing unit 74 compares these computed ortheoretical spectra with actual measured spectra obtained from detectors66, 70.

[0099] It should be noted at this point, that all of the aboveapproaches can be applied to unpolarized light, s-polarized light andp-polarized light. A person skilled in the art will also recognize thatthe method of invention permits one to perform computations andmeasurements for a wide variety of feature geometries and materials onsemi-transparent and transparent substrates. The below selected examplesserve to further illustrate how the method and apparatus of inventionare applied for performing measurements on specific samples.

EXAMPLES

[0100]FIG. 3 illustrates a fused silica sample 99 having adjacentfeatures 100, 102 on a substrate 104. Sample 99 is examined with the aidof apparatus 50 shown in FIG. 2. A beam 106 of incident light 58spanning wavelength range Δλ and originating from source 56 is shown.The remainder of apparatus 50 and response light are not shown in FIG. 3for reasons of clarity.

[0101] Substrate 104 is made of fused silica and features 100 are mesasof fused silica. Features 102 are trenches or air gaps between mesas100. Trenches 102 can be etched or produced in accordance with anysuitable method known in the art.

[0102] In sample 99 material 1 is fused silica and material 2 is air.The area fractions a₁ and a₂ of mesas 100 and trenches 102 are equal andthe depth t_(s) of trenches 102 is 240.9 nm. The calculated responsespectrum of response light (reflected and transmitted light) includesboth the reflectance and transmittance spectra R, T obtained by usingequations 12 and 13 and plotted in FIG. 4. Reflectance R from front(etched) side a (solid line) and back side b (dashed line) of sample 99are referenced by 108 and 110 respectively. Transmittance is drawn insolid line indicated by reference number 112. Reflectance 110 from backside b exhibits more oscillations (peaks and valleys). This is becausefused silica has a higher refractive index than air. This makes itadvantageous to measure reflectance R from the back side b of etchedsample 99. This is especially useful when Cr is coated on sample 99,such as when producing an AAPSM mask.

[0103] The phase φ is calculated using equation 15, and the results areshown by graph 114 in FIG. 5. The phase shift is 180.0 degrees at awavelength λ=248 nm. The destructive interference results in zerointensity in transmittance spectrum 112 at 248 nm as can be seen in FIG.4, since the area fractions a₁ and a₂ are 50%.

[0104] The wavelength for the 180 degree phase shift can be directlymeasured using transmittance spectrum 112, as more clearly shown in FIG.6. In fact, the solid lines are the raw transmittance spectra 112A,112B, 112C, 112D and 112E for five different area fractions a₂ (0%, 5%,10%, 20% and 40%) of trenches 102 in samples analogous to sample 99. Thepositions of the dips are at 248 nm for higher area fractions a₂ andslightly off for lower area fractions a₂. The dashed lines are thetransmittances normalized by the 0% raw transmittance spectrum 112 toremove the wavelength dependence of the substrate 104 spectrum.

[0105] With the aid of normalization the dip position is fixed at 248nm, independent of fractional area a₂ covered by trenches 102. Thisallows one to measure the physical parameter of depth t_(s) of trenches102 and phase shift φ_(T) at any wavelength λ through normalizedtransmittance (by dividing T₁ (=T₂) on both sides of equation 13):

T _(n) =a ₁ ² +a ₂ ²+2a ₁ a ₂ cos φ_(T),  Eq. 29 $\begin{matrix}{{\varphi_{T} = \frac{2{\pi \left( {{n\quad \cos \quad \theta_{2}} - {\cos \quad \theta_{1}}} \right)}t_{s}}{\lambda}},} & {{Eq}.\quad 30}\end{matrix}$

[0106] where n is the refractive index of fused silica, θ₁ is theincident angle and angle θ₂ is given by Snell's law (n sinθ₂=sin θ₁assuming sample 99 is surrounded by air with n_(air)=1) and subscript Ton phase φ indicates that the response light is transmitted. In FIG. 6,θ₁=θ₂=0, and φ_(T)=180° at λ=248 nm. Using n=1.5148 at 248 nm, in themeasurement the measured physical parameter of trench depth of trenches102 is t_(s)=240.9 nm. This result is in excellent agreement with theactual trench depth (240.9 nm) and illustrates the efficacy and accuracyof the method of invention. Once the trench depth is obtained, the phaseshift φ_(T) at any wavelength λ within range Δλ can be calculated byusing equation 30 and the corresponding value of refractive index n. Itshould be noted that T_(n) may have multiple minima when t_(s) is large.

[0107] Similarly, one can obtain the normalized reflectance spectrum R₁on both sides of equation 12:

R _(n) =a ₁ ² +a ₂ ²+2a ₁ a ₂ cos φ_(R).  Eq. 31

[0108] When incident light 58 is illuminated from the back side (sideb), φ_(R) is simply given by: $\begin{matrix}{{\varphi_{R} = \frac{4\pi \quad n\quad t_{s}\cos \quad \theta_{2}}{\lambda}},} & {{Eq}.\quad 32}\end{matrix}$

[0109] where the subscript R on phase φ indicates that the responselight is reflected.

[0110] An example is shown in the graph of FIG. 7 for samples analogousto sample 99 as described above for the same area fractions a₂ asstudied in FIG. 6 (namely 0%, 5%, 10%, 20% and 40%). The peak and valleypositions in the normalized reflectance spectra R_(n) 110A, 110B, 110C,110D and 110E taken from back side b remain constant for correspondingarea fractions a₂ of 0%, 5%, 10%, 20% and 40%. The extremes (peaks andvalleys) are found at φ_(R)=mπ, where m is the interference order and isan even number for peaks and an odd number for valleys. Using equations31 and 32, one can fit R_(n) by varying t_(s) and area fraction a₂ withconstraints from equation 11. Once t_(s) is obtained, one can calculatethe phase shift for any wavelength λ using equation 32. One can alsoobtain t_(s) from two data points in the reflectance spectrum R_(n). Forexample, one can select two extremes (at λ₁ and at λ₂), and calculatet_(s) as follows: $\begin{matrix}{{t_{s} = \frac{\Delta \quad m\quad \lambda_{1}\lambda_{2}}{4{{{n_{1}\lambda_{2}} - {n_{2}\lambda_{1}}}}\cos \quad \theta_{2}}},} & {{Eq}.\quad 33}\end{matrix}$

[0111] where n₁ and n₂ are the refractive indices of substrate 104 atwavelengths λ₁ and λ₂, respectively, and Δm is the order difference ofthe interferences. In FIG. 7 one can choose λ₁=212 nm (valley) andλ₂=699 nm (peak), with n₁=1.539, n₂=1.450, Δm=5, and θ₂=0. With theseparameters equation 33 yields the physical parameter of trench depth oft_(s)=241.0 nm, which is very close to the true value of 240.9 nm. Onceagain, this attests to the efficacy and accuracy of the method andapparatus of the invention.

[0112] When beam 106 of incident light 58 is focused and light 58 isincident from side a, φ_(R) for thick substrate 104 is given by:$\begin{matrix}{\varphi_{R} = {\frac{4\pi \quad n\quad t_{s}\cos \quad \theta_{2}}{\lambda}.}} & {{Eq}.\quad 34}\end{matrix}$

[0113] In this case phase φ_(R) is independent of the refractive index.Hence, one can fit equations 31 and 34 by simply adjusting a₁ ², a₂ ²and t_(s). One can also use equation 33 to calculate t_(s), withn₁=n₂=1.0. Graphs of normalized reflectances for incident light 58 beingilluminated from side a and area fractions a₂ ranging from 0% to 40% asabove are shown in FIG. 8. Note that incident light 58 is focused onfront side a in this case.

[0114]FIG. 9 shows a portion of another sample 120 having a substrate122 of fused silica with trenches 124 (only one shown in FIG. 9). Insample 120 substrate 122 has three films 126A, 126B and 126C on side a.Features 128 are mesas between trenches 124 passing down through allthree films 126A, 126B and 126C. The depth of trenches 124 can bemeasured from front side a or from back side b using equations 21 and22. In general, the measurement from back side b is more sensitive tosubstrate 122 recess. This is particularly true for samples in whichthere is a metal film. For example, a sample 130 with a layer 132 of Cron side a of a fused silica substrate 134 as shown in FIG. 10 is bestexamined from back side b.

[0115]FIG. 11 illustrates the graphs for reflectance measurements onsample 130 from front side a and from back side b. The thickness oflayer 132 of Cr and recess in fused silica were held constant at a totalvalue of 300.9 nm. The recess itself was tested at two values: 240.9 nm(solid line) and 230.9 nm (dashed line), respectively. The measurementfrom front side a shows no sensitivity to the change of recess whereasthe measurement from back side b shows great sensitivity.

ALTERNATIVE EMBODIMENTS

[0116] The present method and apparatus are superior to prior artsolutions, such as the standard transmission-type interferometricapparatus 140 shown in FIG. 12 for comparison purposes. Specifically,apparatus 140 is not convenient for examining a substrate 142 withfeatures that include mesas 144 and air gaps 146. In this case mesashave three layers including at the bottom a layer of the substratematerial, on top of which are located layers 148 and 150. Aninterferometer 152 is positioned to receive two beams 154, 156 passingthrough two adjacent features 144, 146.

[0117] The limitations of interferometric apparatus 140 are, amongother, the fact that beams 154, 156 have to be transmitted throughsubstrate 140 and features 144, 146 in order to enable measurement ofphysical parameters of features 144, 146. When one of layers 148, 150 orsubstrate 142 are not transparent, then apparatus 140 will not be ableto perform the measurement.

[0118] In contrast, an apparatus according to the present inventionoffers a number of options for measuring one or more physical parametersof features 144, 146 on substrate 142 irrespective of thetransmissibility of layers 148, 150. As shown in FIG. 13 this can beperformed in the reflective mode using a beam of incident light 160 anda reflected response light 162. Alternatively, a beam of incident light164 illuminates substrate 142 from the back side to produce a reflectedresponse light 166, as shown in FIG. 14. As noted in the examples above,under certain circumstances measurement using reflected response light166 from the back side will provide higher accuracy measurements thanlight 162 reflected from the front side. In still another embodiment, abeam of incident light 168 is directed at substrate 142 and atransmitted response light 170 is measured. As will be appreciated by aperson skilled in the art, a combination of all three methodsillustrated in FIGS. 13-15 can be used depending on the transmission andreflection properties of substrate 142 and features 144, 146.

[0119]FIG. 16 illustrates a portion of yet another apparatus 180 forexamining physical parameters of features 182, 184 of a sample 186. Inthis embodiment features 182 are embedded within feature 184. Feature184 is a flat film deposited on a semi-transparent substrate 188.Apparatus 180 has a source 190 for producing a beam of incident light192 for illuminating sample 186. Apparatus 180 has a detector 194 forexamining a response light 196 reflected by sample 186. It should benoted that a transmitted response light 198 can also be measured asnecessary. Transmitted response light 198 can be measured by detector194 with the aid of additional optics (not shown) or a separate detectorand optics (not shown).

[0120] Apparatus 180 takes advantage of the fact that refractive,catadioptric or purely reflective optics can be used to guide incidentlight 192 and response light 196 (198). In fact, purely reflectiveoptics are advantageous when incident wavelength range Δλ is large,e.g., when it extends from 190 nm to 1000 nm. In the present embodimentΔλ is large and thus apparatus 180 employs a set of reflective optics200, 202 in the form of curved mirrors. Mirror 200 directs incidentlight 192 to sample 186. Mirror 202 receives response light 196 fromsample 186 and directs it to detector 194. In a preferred version ofapparatus 180 mirrors 200, 202 are torroidal mirrors. For generalinformation about the use of torroidal mirrors the reader is referred toU.S. Pat. No. 5,991,022.

[0121] In view of the above, it will be clear to one skilled in the artthat the above embodiments may be altered in many ways without departingfrom the scope of the invention. Accordingly, the scope of the inventionshould be determined by the following claims and their legalequivalents.

What is claimed is:
 1. A method for determining a physical parameter offeatures on a substrate, said method comprising: a) illuminating saidsubstrate with an incident light having an incident wavelength range Δλwithin which said substrate is at least semi-transparent such that saidincident light enters said substrate and said features; b) receiving aresponse light from said substrate and said features; c) measuring aresponse spectrum of said response light; d) computing a complex-valuedresponse due to said features and said substrate; e) determining saidphysical parameter from said response spectrum and said complex-valuedresponse.
 2. The method of claim 1, wherein said complex-valued responsecomprises at least one response selected from the group consisting ofcomplex reflectance amplitude and complex transmittance amplitude andsaid response spectrum comprises at least one response spectrum selectedfrom the group consisting of reflectance R and transmittance T.
 3. Themethod of claim 2, wherein said at least one complex-valued response isa complex reflectance amplitude and said method further comprisescomputing said reflectance R by multiplying said complex reflectanceamplitude with its complex conjugate.
 4. The method of claim 2, whereinsaid at least one complex-valued response is a complex transmittanceamplitude and said method further comprises computing said transmittanceT by multiplying said complex transmittance amplitude with its complexconjugate.
 5. The method of claim 1, wherein said incident light has avertical coherence length L_(vc) sufficiently small with respect to athickness d_(s) of said substrate to produce incoherence in saidresponse light.
 6. The method of claim 5, further comprising averaging aphase δ_(s) of said complex-valued response.
 7. The method of claim 1,wherein said incident wavelength range Δλ is selected such that saidfeatures produce a coherent fraction β and an incoherent fraction (1−β)in said response light.
 8. The method of claim 7, further comprisingdetermining said coherent fraction β from a lateral coherence lengthL_(lc) of said incident light.
 9. The method of claim 1, wherein saidfeatures are periodic and said incident light is focused to anillumination area covering a sufficiently small number of said featuressuch that diffraction effects are negligible.
 10. The method of claim 1,wherein an area of said features is larger than a lateral coherencelength L_(lc) of said incident light and said method comprisesincoherently adding said complex-valued response from said features. 11.The method of claim 1, wherein an area of at least one of said featuresis smaller than a lateral coherence length L_(lc) of said incident lightand said method comprises coherently adding said complex-valued responsefrom said features.
 12. The method of claim 11, wherein said featurescomprise adjacent features made of a material 1 and covering a firstarea fraction a₁ illuminated by said incident light and of a material 2covering a second area fraction a₂ illuminated by said incident light,and wherein coherently adding said complex-valued response comprisescoherently adding at least one complex-valued response selected from thegroup consisting of a total complex-valued reflectance amplitude r_(C)and a total complex-valued transmittance amplitude t_(C).
 13. The methodof claim 12, wherein said total complex-valued reflectance amplituder_(C) is computed as: r _(C) =a ₁ r ₁ +a ₂ r ₂, and said totalcomplex-valued transmittance amplitude t_(C) is computed as: t _(C) =a ₁t ₁ +a ₂ t ₂, and where a₁+a₂=1.
 14. The method of claim 12, whereinsaid response spectrum comprises at least one response spectrum selectedfrom the group consisting of a coherent reflectance R_(C) and a coherenttransmittance T_(C).
 15. The method of claim 14, wherein said coherentreflectance R_(C) is calculated by using a cross term:${{\langle{r_{1} \cdot r_{2}^{*}}\rangle} = \frac{{r_{1,{as}}r_{2,{as}}^{*}} + {\left( {{t_{1,{as}}t_{2,{as}}^{*}t_{1,{sa}}t_{2,{sa}}^{*}} - {r_{1,{as}}r_{2,{as}}^{*}r_{2,{sa}}r_{2,{sa}}^{*}}} \right)r_{1,{sb}}r_{2,{ab}}^{*}^{{- 2}\alpha_{s}d_{s}}}}{1 - {r_{1,{sa}}r_{2,{sa}}^{*}r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}}},$

where α_(s) is an absorption coefficient of said substrate and d_(s) isa thickness of said substrate.
 16. The method of claim 15, wherein saidincident light is focused on a back side of said substrate and saidcross term is computed as

r ₁ ·r ₂

*

=t _(1,as) t _(2,as) *t _(1,sa) t _(2,sa) *r _(1,sb) r _(2,sb) *e ^(−2α)^(_(s)) ^(d) ^(_(s.))
 17. The method of claim 14, wherein said coherenttransmittance T_(C) is calculated by using a cross term:${{\langle{t_{1} \cdot t_{2}^{*}}\rangle} = {\frac{t_{1,{as}}t_{2,{as}}^{*}t_{1,{sb}}t_{2,{sb}}^{*}^{{- \alpha_{s}}d_{s}}}{1 - {r_{1,{sa}}r_{2,{sa}}^{*}r_{1,{sb}}r_{2,{sb}}^{*}^{{- 2}\alpha_{s}d_{s}}}} = {A\quad ^{\quad \varphi}}}},$

where A is the amplitude of

t₁·t₂*

, φ is the phase shift between t₁ and t₂, α_(s) is an absorptioncoefficient of said substrate and d_(s) is a thickness of saidsubstrate.
 18. The method of claim 1, further comprising collimatingsaid incident light.
 19. The method of claim 1, further comprisingfocusing said incident light.
 20. The method of claim 19, wherein saidincident light is focused on a surface of said substrate.
 21. The methodof claim 1, wherein said physical parameter is selected from the groupconsisting of depth, width, real part of the complex refractive index,imaginary part of the complex refractive index.
 22. The method of claim1, further comprising computing a phase shift δ in said complex-valuedresponse.
 23. The method of claim 1, wherein said at least one featureis positioned on a first side of said substrate and said methodcomprises illuminating said substrate from a side opposite said firstside and focusing said incident light.
 24. The method of claim 1,wherein said at least one feature is positioned on a first side of saidsubstrate and said method comprises illuminating said substrate withsaid incident light from said first side.
 25. The method of claim 1,wherein said physical parameter is derived from a phase shift φ in saidresponse light.
 26. The method of claim 25, wherein said response lightis transmitted such that:$\varphi = {\varphi_{T} = {\frac{2{\pi \left( {{n\quad \cos \quad \theta_{2}} - {\cos \quad \theta_{1}}} \right)}t_{s}}{\lambda}.}}$


27. The method of claim 25, wherein said response light is reflectedsuch that:$\varphi = {\varphi_{R} = \frac{4\pi \quad n\quad t_{s}\cos \quad \theta_{2}}{\lambda}}$


28. The method of claim 1, wherein said features comprise at least onefilm deposited on said substrate.
 29. The method of claim 28, whereinsaid features comprise at least one embedded feature in said at leastone film.
 30. An apparatus for determining a physical parameter offeatures on a substrate, said apparatus comprising: a) an illuminationsource for producing an incident light having an incident wavelengthrange Δλ within which said substrate is at least semi-transparent; b)optics for guiding said incident light such that said incident lightenters said substrate and said features; c) a detector for receiving aresponse light from said substrate and said features and measuring ameasured response spectrum of said response light; d) a processing unitfor computing a complex-valued response of said features and saidsubstrate and for determining said physical parameter from said measuredresponse spectrum and said complex-valued response.
 31. The apparatus ofclaim 30, wherein said substrate is transparent within said incidentwavelength range Δλ.
 32. The apparatus of claim 30, wherein saidsubstrate is optically thick.
 33. The apparatus of claim 30, whereinsaid illumination source is broadband and said incident wavelength rangeΔλ extends from about 190 nm to about 1000 nm.
 34. The apparatus ofclaim 30, wherein at least one of said features comprises a metal layer.